Many complex systems - be they financial, natural, or social - are composed of units - such as stocks, neurons, or agents - whose joint activity can be represented as a multivariate time series. An issue of both practical and theoretical importance concerns the possibility of inferring the presence of a static relationship between any two units solely from their dynamic behaviour. The present contribution aims at tackling such an issue within the framework of traditional hypothesis testing: briefly speaking, our suggestion is that of linking any two units if behaving in a sufficiently similar way. To achieve such a goal, we project a multivariate time series onto a signed graph by i) comparing the empirical properties of the former with those expected under a suitable benchmark and ii) linking any two units with a positive (negative) edge in case the corresponding series shares a significantly large number of concordant (discordant) values. To define our benchmarks, we adopt an information-theoretic approach that is rooted into the constrained maximisation of Shannon entropy, a procedure inducing an ensemble of multivariate time series that preserves some of the empirical properties on average, while randomising everything else. We showcase the possible applications of our method by addressing one of the most timely issues in the domain of neurosciences, i.e. that of determining if brain networks are frustrated or not, and, if so, to what extent. As our results suggest, this is indeed the case, with the major contribution to the underlying negative subgraph coming from the subcortical regions (and, to a lesser extent, from the limbic ones). At the mesoscopic level, the minimisation of the Bayesian Information Criterion, instantiated with the Signed Stochastic Block Model, reveals that brain regions gather into modules aligning with the statistical variant of the Relaxed Balance Theory.

翻译：许多复杂系统——无论是金融、自然还是社会系统——均由多个单元（如股票、神经元或智能体）组成，这些单元的联合活动可以表示为多变量时间序列。一个兼具实践与理论重要性的问题在于：能否仅通过各单元的动态行为推断出任意两个单元之间是否存在静态关系。本文旨在传统假设检验框架内解决这一问题：简言之，我们建议将行为足够相似的两个单元建立联系。为实现该目标，我们将多变量时间序列投影到有符号图（signed graph）上，具体步骤如下：i) 比较时间序列的经验属性与在适当基准模型下的预期属性；ii) 若对应序列具有显著大量的一致（不一致）值，则以正（负）边连接这两个单元。为定义基准模型，我们采用基于香农熵约束最大化的信息论方法，该方法生成一个多变量时间序列的集成系统，在平均意义上保留部分经验属性，同时随机化其他所有属性。我们通过解决神经科学领域最前沿的问题之一（即判断脑网络是否处于受挫（frustrated）状态及其受挫程度）来展示本方法的应用潜力。结果表明，脑网络确实存在受挫现象，其中负子图的主要贡献来自皮层下区域（其次为边缘区域）。在中观层面，以有符号随机块模型（Signed Stochastic Block Model）实例化的贝叶斯信息准则最小化揭示，脑区域聚集为与松弛平衡理论（Relaxed Balance Theory）统计变体相一致的模块。